Group #27304999 — subgroups

Subgroups — #27304999

Order 13568.

158 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276992	C₂	1	1	yes	maximal, characteristic
2	Q₃₂×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#20272917	C₂	1	1	yes	maximal, characteristic
4	#20272914	C₂	1	1	yes	maximal, characteristic
5	#20272910	C₂	1	1	yes	maximal, characteristic
6	#20272896	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#1416	C₂	1	1	yes	maximal, characteristic
8	#16811024	—	2	1	no	—
9	Q₁₆×Dic₅₃	—	2	1	no	—
10	C₂×#13133123	C₂²	1	1	yes	characteristic
11	#16810450	—	2	1	no	—
12	#16810449	C₂²	1	1	yes	characteristic
13	#16810449	C₂²	1	1	yes	characteristic
14	#16810427	—	2	1	no	—
15	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
16	C₂×Q₃₂×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#366	C₂²	1	1	yes	characteristic
19	#13133208	—	4	1	no	—
20	Q₈×Dic₅₃	—	4	1	no	—
21	C₂×#11719908	D₄	1	1	yes	characteristic
22	#13133151	—	2	1	no	—
23	#13133151	—	2	1	no	—
24	#13133148	—	4	1	no	—
25	#13133141	—	4	1	no	—
26	C₈×Dic₅₃	D₄	1	1	yes	characteristic
27	#13133123	—	2	1	no	—
28	C₂×Q₁₆×C₅₃	—	2	1	no	—
29	Q₃₂×C₅₃	—	2	1	no	—
30	Q₃₂×C₅₃	Q₈	1	2	yes	—
31	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
32	#107×C₅₃	—	2	1	no	—
33	C₃₂×C₅₃	—	2	1	no	—
34	C₂×Q₈×C₅₃	—	4	1	no	—
35	Q₁₆×C₅₃	—	2	2	no	—
36	Q₁₆×C₅₃	—	4	1	no	—
37	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₄⋊C₄×C₅₃	—	4	1	no	—
39	C₄⋊Dic₅₃	—	4	1	no	—
40	C₄⋊Dic₅₃	—	4	1	no	—
41	C₄⋊Dic₅₃	—	8	1	no	—
42	C₄⋊Dic₅₃	—	8	1	no	—
43	#11719919	—	8	1	no	—
44	C₄×Dic₅₃	—	8	1	no	—
45	C₄×Dic₅₃	D₈	1	1	yes	characteristic
46	C₂×#88322	D₈	1	1	yes	characteristic
47	C₁₆×C₅₃	C₂×Q₈	1	1	yes	characteristic
48	C₁₆×C₅₃	#41	1	1	yes	characteristic
49	#11719908	—	2	1	no	—
50	Q₈×C₅₃	—	4	2	no	—
51	Q₈×C₅₃	—	8	1	no	—
52	C₂×C₄×C₅₃	—	8	1	no	—
53	C₂×C₄×C₅₃	—	8	1	no	—
54	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
55	C₂×Dic₅₃	—	8	1	no	—
56	C₂×Dic₅₃	—	8	1	no	—
57	C₂×Dic₅₃	D₁₆	1	1	yes	characteristic
58	C₂×Dic₅₃	D₁₆	1	1	yes	characteristic
59	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
60	C₈×C₅₃	#137	1	1	yes	characteristic
61	#88322	—	2	1	no	—
62	#13708	—	53	1	no	maximal, Sylow
63	C₂²×C₅₃	C₂×D₁₆	1	1	yes	characteristic
64	C₄×C₅₃	—	8	2	no	—
65	C₄×C₅₃	—	16	1	no	—
66	C₄×C₅₃	#474	1	1	yes	characteristic
67	C₄×C₅₃	#510	1	1	yes	characteristic
68	Dic₅₃	—	2	1	no	—
69	Dic₅₃	—	2	1	no	—
70	Dic₅₃	—	16	1	no	—
71	Dic₅₃	—	16	1	no	—
72	#2253	—	53	1	no	—
73	C₄×Q₃₂	—	53	1	no	—
74	#1424	—	53	1	no	—
75	#1423	—	53	1	no	—
76	C₄⋊C₃₂	—	53	1	no	—
77	#1416	—	53	1	no	—
78	#1416	D₅₃	1	1	yes	characteristic
79	C₂×C₅₃	#2224	1	1	yes	characteristic
80	C₂×C₅₃	#2262	1	1	yes	characteristic
81	C₂×C₅₃	#2264	1	1	yes	characteristic
82	C₂×Q₃₂	C₂×D₅₃	1	1	yes	characteristic
83	#499	—	106	1	no	—
84	C₄×Q₁₆	—	106	1	no	—
85	C₂×C₃₂	—	53	1	no	—
86	C₂×C₃₂	C₂×D₅₃	1	1	yes	characteristic
87	#367	—	106	1	no	—
88	#366	—	53	1	no	—
89	#366	—	53	1	no	—
90	#366	C₂×D₅₃	1	1	yes	characteristic
91	#358	—	106	1	no	—
92	C₄×C₁₆	—	53	1	no	—
93	C₅₃	#13708	1	1	yes	characteristic, complemented, Sylow
94	C₂×Q₁₆	—	2	1	no	—
95	#125	—	212	1	no	—
96	C₄×Q₈	—	212	1	no	—
97	Q₃₂	—	2	1	no	—
98	Q₃₂	Dic₁₀₆	1	2	yes	—
99	C₂×C₁₆	—	53	1	no	—
100	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
101	#107	—	2	1	no	—
102	#107	—	106	1	no	—
103	#107	—	106	1	no	—
104	#106	—	212	1	no	—
105	#103	—	212	1	no	—
106	C₄×C₈	—	53	1	no	—
107	C₃₂	—	2	1	no	—
108	C₃₂	—	106	1	no	—
109	C₂×Q₈	—	4	1	no	—
110	Q₁₆	—	2	2	no	—
111	Q₁₆	—	4	1	no	—
112	C₂×C₈	—	53	1	no	—
113	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
114	C₄⋊C₄	—	4	1	no	—
115	C₄⋊C₄	—	424	1	no	—
116	C₄⋊C₄	—	424	1	no	—
117	C₄⋊C₄	—	424	1	no	—
118	C₄⋊C₄	—	212	1	no	—
119	C₄⋊C₄	—	212	1	no	—
120	C₄²	—	53	1	no	—
121	C₄²	—	424	1	no	—
122	C₁₆	—	106	1	no	—
123	C₁₆	C₂×Dic₁₀₆	1	1	yes	characteristic
124	C₁₆	#11719947	1	1	yes	characteristic
125	Q₈	—	4	2	no	—
126	Q₈	—	8	1	no	—
127	C₂×C₄	—	8	1	no	—
128	C₂×C₄	—	8	1	no	—
129	C₂×C₄	—	53	1	no	—
130	C₂×C₄	—	53	1	no	—
131	C₂×C₄	—	424	1	no	—
132	C₂×C₄	—	424	1	no	—
133	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
134	C₈	—	106	1	no	—
135	C₈	#13133194	1	1	yes	characteristic
136	C₈	#13133246	1	1	yes	characteristic
137	C₂²	D₁₆×D₅₃	1	1	yes	characteristic
138	C₄	—	8	2	no	—
139	C₄	—	16	1	no	—
140	C₄	—	848	1	no	—
141	C₄	—	848	1	no	—
142	C₄	—	106	1	no	—
143	C₄	—	106	1	no	—
144	C₄	#16810901	1	1	yes	characteristic
145	C₄	#16811062	1	1	yes	characteristic
146	C₂	#20276850	1	1	yes	characteristic
147	C₂	#20277023	1	1	yes	characteristic
148	C₂	#20277034	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276992

C₂

yes

maximal, characteristic

Q₃₂×Dic₅₃

C₂

yes

maximal, characteristic

#20272917

C₂

yes

maximal, characteristic

#20272914

C₂

yes

maximal, characteristic

#20272910

C₂

yes

maximal, characteristic

#20272896

C₂

yes

maximal, characteristic

C₅₃×#1416

C₂

yes

maximal, characteristic

#16811024

—

Q₁₆×Dic₅₃

—

C₂×#13133123

C₂²

yes

characteristic

#16810450

—

#16810449

C₂²

yes

characteristic

#16810449

C₂²

yes

characteristic

#16810427

—

C₁₆×Dic₅₃

C₂²

yes

characteristic

C₂×Q₃₂×C₅₃

C₂²

yes

characteristic

C₂×C₃₂×C₅₃

C₂²

yes

characteristic

C₅₃×#366

C₂²

yes

characteristic

#13133208

—

Q₈×Dic₅₃

—

C₂×#11719908

D₄

yes

—

—

—

—

C₈×Dic₅₃

D₄

yes

characteristic

#13133123

—

C₂×Q₁₆×C₅₃

—

Q₃₂×C₅₃

—

Q₃₂×C₅₃

Q₈

yes

—

C₂×C₁₆×C₅₃

C₂³

yes

characteristic

#107×C₅₃

—

C₃₂×C₅₃

—

C₂×Q₈×C₅₃

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

D₈

yes

characteristic

C₂×#88322

D₈

yes

characteristic

C₁₆×C₅₃

C₂×Q₈

yes

C₁₆×C₅₃

yes

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

D₁₆

yes

characteristic

C₂×Dic₅₃

D₁₆

yes

characteristic

C₈×C₅₃

C₂²⋊Q₈

yes

C₈×C₅₃

yes

—

—

C₂²×C₅₃

C₂×D₁₆

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

#2253

—

C₄×Q₃₂

—

#1424

—

#1423

—

C₄⋊C₃₂

—

#1416

—

#1416

D₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×Q₃₂

C₂×D₅₃

yes

characteristic

#499

—

106

—

C₄×Q₁₆

—

106

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₅₃

yes

characteristic

#367

—

106

—

#366

—

#366

—

#366

C₂×D₅₃

yes

characteristic

#358

—

106

—

C₄×C₁₆

—

C₅₃

#13708

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

#125

—

212

—

C₄×Q₈

—

212

—

Q₃₂

—

Q₃₂

Dic₁₀₆

yes

—

C₂×C₁₆

—

100

C₂×C₁₆

C₂²×D₅₃

yes

characteristic

101

#107

—

102

#107

—

106

—

103

#107

—

106

—

104

#106

—

212

—

105

#103

—

212

—

106

C₄×C₈

—

107

C₃₂

—

108

C₃₂

—

106

—

109

C₂×Q₈

—

110

Q₁₆

—

111

Q₁₆

—

112

C₂×C₈

—

113

C₂×C₈

D₄×D₅₃

yes

characteristic

114

C₄⋊C₄

—

115

C₄⋊C₄

—

424

—

116

C₄⋊C₄

—

424

—

117

C₄⋊C₄

—

424

—

118

C₄⋊C₄

—

212

—

119

C₄⋊C₄

—

212

—

120

C₄²

—

121

C₄²

—

424

—

122

C₁₆

—

106

—

123

C₁₆

C₂×Dic₁₀₆

yes

characteristic

124

C₁₆

#11719947

yes

characteristic

125

Q₈

—

126

Q₈

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

424

—

132

C₂×C₄

—

424

—

133

C₂×C₄

D₈×D₅₃

yes

characteristic

134

C₈

—

106

—

135

C₈

#13133194

yes

characteristic

136

C₈

#13133246

yes

characteristic

137

C₂²

D₁₆×D₅₃

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

848

—

141

C₄

—

848

—

142

C₄

—

106

—

143

C₄

—

106

—

144

C₄

#16810901

yes

characteristic

145

C₄

#16811062

yes

characteristic

146

C₂

#20276850

yes

characteristic

147

C₂

#20277023

yes

characteristic

148

C₂

#20277034

yes

characteristic

	subgroup	G/H
center	C₂²	D₁₆×D₅₃
commutator, cosocle	C₂×C₁₆×C₅₃	C₂³
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting	C₅₃×#1416	C₂
socle, prime radical	C₂²×C₅₃	C₂×D₁₆
hypercenter	#1416	D₅₃
hypocenter	C₅₃	#13708
LCS	G ▷ C₂×C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1416

subgroup

G/H

center

C₂²

D₁₆×D₅₃

commutator, cosocle

C₂×C₁₆×C₅₃

C₂³

Frattini

C₂×C₁₆

C₂²×D₅₃

Fitting

C₅₃×#1416

C₂

socle, prime radical

C₂²×C₅₃

C₂×D₁₆

D₅₃

C₅₃

G ▷ C₂×C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1416

N	H ≅ G/N	type
93. C₅₃	#13708	semidirect

H ≅ G/N

type

93. C₅₃

#13708

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20276992	2
Q₃₂×Dic₅₃	2
#20272917	2
#20272914	2
#20272910	2
#20272896	2
C₅₃×#1416	2
#13708	53

subgroup

index

Q₃₂×Dic₅₃

C₅₃×#1416

Subgroups — #27304999

Conjugacy classes of subgroups

Complements

Maximal subgroups